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Friday, July 2, 2010

Union

This summer I am very purposefully avoiding thinking about work. A break seems crucial and necessary. I can't really explain why. I can't think about work. I don't want to think about work. My brain and body rebel at the notion of thinking about work. When people on Twitter start talking about the minutiae of their grading systems, I have to close the laptop and leave the room. This is the first time I've begun summer feeling like this. Maybe I will understand it better when I come out the other side.

One of the things I've been doing instead is attending Bikram yoga practice. The executive summary of Bikram is: very hot room, an hour and a half, same difficult 26 poses in the same order every time, lots of sweat. I've gone every day. I am shooting for thirty days in a row. My everything hurts.

But as you've probably guessed, despite my determination to think about other things for six weeks, aphorisms from Bikram apply without editing to a teaching practice. This is just me writing some of them down.

Everything matters. At yoga, which direction your fingers are pointing and where your eyes are focused matter. At school, where you stand, how long you pause, and the numbers you choose for every problem matter.

...but don't be too serious about it. Wink at yourself in the mirror.

Many teachers is better than one teacher. At yoga, I haven't had the same instructor twice. They lead you through the same poses, but the individuals are all different and equally awesome. This one told me to point my tailbone at the floor so that I really felt my spine lengthen, that one told me to press my chin into my chest. At school you can and should engage all the students in helping teach the course. This goes to deep, philosophical methods by which you approach instruction with collaborative problem solving, and the surface of how you structure practice activities.

Push yourself, try your best, and aim for perfection. At yoga, you can move a half inch deeper into the pose on the next breath. You can inhale another sip of air when your lungs are full up. At school, you can pick one student in each class you haven't talked to this week and ask them about their sport/hobby/pet.

...but be gentle and forgiving, and kinder to yourself than you think you deserve.

...and then let it go. Did you fall out of standing bow? Twice? It's over now. Let it go. One of the instructors says this and it's awesome: "Exhale...set you free." Did something go down at school you could have handled better? Acknowledge, learn, let it go.

If you are doing something mentally and physically demanding, don't forget to eat and drink enough water. Or you will feel like crap. At school, sometimes I am feeling cranky in the late afternoon and realize I haven't had any water all day. The consequences are a bit more extreme at Bikram: dizziness, nausea, feeling faint.

Finally, I took a picture of their poster, which might make a cute WCYDWT. What are they trying to maximize or promote with this pricing scheme? You'd probably want to hide that bottom part at first.

How clearly we see things when we aren't looking. I believe that NOT thinking about teaching after a long year is the best thing you can do as a teacher. It's all about perspective. You have none when you are on the inside. By stepping outside of it all at the end of the year we are able to (eventually) look back on everything with a new perspective, and more objectively.

The same thing happens each and every time I go run. I tell myself I am going to let it all go and not think about work. I turn the ipod all the way up. I run fast and hard. And yet, this is when my best ideas come to me. These are often my "eureka" moments. As I see it, you are opening up yourself to a summer of "eureka".

I think it is so interesting how we teachers can apply ANYTHING to teaching. Is this common in other professions or just a teaching obsession? I've heard before that teaching is not a career but a calling. And sometimes I am forced to believe that since we can't escape it. Even in summer.

Very zen. Actually seems like a good way of helping students *learn* math, as in: it's just exercise for the brain. Stretching. No right answers. It's all process, it's all path towards some higher understanding, however one defines "height." At some point in human history, Aristotle codified the rules of logic (see: Organon), and mandated the Law of the Excluded Middle, which in a nutshell says that 1 cannot equal 0. Maybe that's true. Maybe not. For the Hindu mathematicians, the words for "zero" and "infinity" were both the same, but who knows.

It doesn't really matter, anyway. What matters is that if math itself isn't necessarily "right," then we--teachers, students, whoever--don't really need to worry so much about whether we're doing it correctly, for the "it" is now something much deeper, more transcendental. Breath in. Breath out. No worries.

To Kate's point about trying to escape teaching, but not being able...a poetry teacher in college and I have a years-long debate about what makes a good poem. I say intention. If the poet intends something beautiful--if the poem is authentic--then the writing is by definition good. He disagrees. He says that poetry--all art, really--involves more than that: that it must definitely involve the creation itself. In the end, we compromise: art is an ethic. It involves unwavering intentionality, but also a heightened manipulation of the scalpel. Maybe teaching's the same way. It involves meta-cognition--"where should I stand? How should I respond?"--wait time, green or red?, etc, etc, etc--but also a less tangible sense of Teacherness. Like art, it's an ethic.

You find yourself in your 30s, you wake up, you look in the mirror and say, "I'm a teacher. It's what I am." And that self-awareness doesn't stop in the summer, any more than does being a parent (I assume).

It's July. Spain just beat Paraguay. I don't know about you guys, but it's 80 degrees and perfect here in DC. But here we are: reading & commenting on teacher blogs. July? Yeah, what about it?

the Law of the Excluded Middle, which in a nutshell says that 1 cannot equal 0.

It says nothing of the kind. I know that all sorts of mumbo-jumbo is considered intellectually avant-garde, especially if it advances a radically relativist position, but at least get your facts right.

To which point: math itself is necessarily right, and there is such a thing as correctness. The LEM is a particular concept and just because you're confused about it doesn't mean the concept itself is confused; it just means you're wrong.

In fact, mathematics -- in and of itself -- is concerned only with concepts which are so clean-cut. And if you're teaching students anything else then remind me not to drive over any bridges they design.

You make important points, unapologetic, and entirely apropos ones. However,I'd have to disagree with your premise that math is clean cut, anddeals only with concepts that we understand. In fact, math is exactlythe thing we invented to try to get our heads around things that wedon't understand. Our troubled relationship with irrational numbers?Infinity? Zero? This isn't intellectual, post-modern relativism.It's historical fact. And the evolution of math closely mirrors ourown evolving understanding of reality and truth. One day irrationalnumbers don't exist. The next day they do. One day nature abhors avacuum. The next day it's the fulcrum of our number line. Realitychanges.

That said, you're right: there are aspects of mathematics that are*true*, where we define truth by what seems to work. I wouldn't wantto be a passenger on a jet built using non-dual mathematics, either, thoughnor would I--and more to the point, nor do we--live in a world wherethis is the *only* path of logic/mathematics. Again, mathematics ishistorically as much a philosophy as anything else. It's not mesaying that, either. Just ask Aristotle or, for that matter, Hippasus(Cantor, Gödel, etc).

Still, the more important question is how we weave this version ofmath into the narrative that we're supposed to teach, which is to say,how we help define "math" to our students. If you start by saying,"This all may or may not be true," it may make for an interestingfirst day, but no doubt a tough and fairly pointless year. At thesame time, if we define it as a clean-cut collection of operations,rather than a way of thinking (and unthinking, rethinking...), thenwe're effectively defining architecture as the ruler, not to mentionrobbing students of a profoundly beautiful "other."

With respect to the Law of the Excluded Middle, then--in which Aristotle helped to codify logic--Aristotle says,"...it will not be possible to be and not to be the same thing," whichis indeed saying that 1(be) cannot equal zero (not be). Of course,you've no doubt seen the whole:

Let a = b = 1a^2 - b^2 = a^2 - ab(a+b)(a-b) = a(a-b)a+b = ab = 0

Yes, the *trick* is division by zero, and mathematicians better than Iwill explain why it's illegal, but ultimately it comes down tosomething pretty simple: the result doesn't "make sense," and so weforbade it. It's the one law we agreed not to break. And good thing,too. To your point, a bridge where 1=0 makes for petty high insurancepremiums. But all I'm saying is, let's not forget that this thingthat we decided to teach wasn't some remnant of the Big Bang. Weinvented math, and its *truth* is only as good as our ability todiscern it...an ability which is in a constant state of flux, which Isuspect is exactly what makes this whole pursuit interesting.

Anyway, perhaps you'd agree that talking about "right" and "wrong" inthis context is fairly circular. Ultimately, I entirely agree withyou, but for a slightly different reason. Relativism isn'tfrustrating because it's not true (that's like telling an atheist thathe's going to hell), but because it doesn't allow for goals. Ifeverything is equally good, then what's there to aspire to? In thiscase, ignuranse is as virtuous as wisdom, which seems fairly absurd.At the same time, it doesn't mean that it's fundamentally wrong,either. Is math right? Is math wrong?

Maybe a better question is, Is math complete? Does 1=0? I don't know. And that's the point.

It doesn't mean we don't study mathematics. Rather, it demands that we do. Neo had to master the Matrix first, right?

You're confused. The LEM doesn't say that it's impossible for something to be both true and false, it says that it's impossible to be neither. Anything that's not true is false, and vice versa.

Words have meanings -- thought you use enough of them that maybe they get lost somewhere. The point of the law of the excludedmiddle is that it excludes the possibility that there's anything in the middle of true and false.

Maybe you don't know whether 1=0, but I do. When discussing any sort of number that shows up in high school classes, the answer is an unequivocal "no". It's just not up for debate. And it's not just because of some social convention; mathematics is as much a part of natural law as physics is, if not more so. To dip into your florid idiom: the integers were around before the big bang.

I'm sorry this didn't occur to me earlier, but the rightness or wrongness of math--incompleteness, completeness, inverse operations, etc.--isn't really the point. This isn't a blog about mathematics per se, after all, but rather the teaching of it.

Unapologetic: you say you wouldn't want your students to be in a class where the teacher claimed that 1=0. No doubt. But would you want them to be in a class where the teacher seemed to hold some secret in her hands, and demanded, "I hold The Truth, and whoever disagrees is wrong?"

I visited your website. You clearly have a deep reverence and passion for the inquiry that is mathematics--for the process--and that's something to be respected and encouraged. But so, too, is Socratic humility. As teachers, we have to model that, lest we lead our students off the edge of our flat earth. For if we continue to claim that there's only one way to some truth--namely, ours--then we risk losing a sizable majority of the next generation, and the next, only to ensure the perpetuation of the mantra, "I don't do math."

Anyway, I know your reaction was to my initial post, which was inarticulate at best. (Sorry about that). In the end, maybe Kate said it best with respect to effective math teaching: "don't be so serious about it."

You're not wrong. But that doesn't mean you're right, either. The middle. Also called: nuance.

You seem certain about what's known. You also seem certain about what's not known.

Done respectfully, this could be an interesting discussion, and I'm happy to continue it with you one-on-one (karimkai@gmail.com). However, I think Kate was aspiring to something a bit nicer than this dialogue has become.

It's possible to encourage the process without saying -- and I quote -- "no right answers". There are right answers, and the process is useless if it doesn't lead there. If a student has a very creative, idiomatic approach that leads her to conclude that 1=0, then what good is it?

Fair point. I think we're talking about two different things. If either of us asks a student to solve 2x+3=7, of course we're only going to accept x=2. In this case, there's clearly a "right" answer, and I'm sorry if my initial post seemed to point to some post-modern, kumbaya approach to mathematics. Indeed, that wouldn't be very useful at all, which is your point.

I just ran into your blog, and I love it! I just started a website called "TeacherThink" which challenges educators to innovate by providing how-tos, reviews, and a general place for teacher muse. You can find it @ www.TeacherThink.com. I am also on twitter @ TeacherThink.